- Email: [email protected]

Jeongmin Lee‡

January 4, 2017

Abstract This paper investigates the strategic foundations for rational expectations equilibrium. In the model, risk-averse traders with two signal—private information and endowment shocks—submit demand schedules to trade a risky asset. Traders are divided into groups. Within groups, traders share common signals; signals are different across different groups. Either traders become price takers or the price becomes fully revealing, but not both, as the number of competitors per group goes to infinity. As the number of groups goes to infinity, neither price taking nor fully revealing prices are obtained. Measuring competition by the quantity traded as a fraction of that traded by a price-taker, we show that optimal exercise of market power has opposite implications for competition and price informativeness. Keywords: Rational expectations equilibrium, private information, endowment shocks, price informativeness, strategic trading, competition, noise trading.

* We

thank Leland Crane, Julien Cujean, Jason Donaldson, Phil Dybvig, Maryam Farboodi, Jungsuk Han, Tibor Heumann, Ohad Kadan, Mark Loewenstein, Rich Mathews, Stephen Morris, Anna Obizhaeva, Giorgia Piacentino, Marzena Rostek, Shri Santosh, and Yajun Wang for helpful comments. † Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA; [email protected] Kyle has worked as a consultant for various companies, exchanges, and government agencies. He is a non-executive director of a U.S.-based asset management company. ‡ Olin Business School, Washington University, St. Louis, MO 63130, USA; [email protected]

The perfect competition assumption, widely used in many areas of economics, assumes agents are so small that they take prices as given. By applying this assumption to information asymmetry, rational expectations equilibrium features both price-taking behavior and fully revealing prices. For example, Grossman and Stiglitz (1980) and Hellwig (1980) assume perfect competition and show the price becomes fully revealing as noise trading vanishes. This is problematic because traders must ignore their effect of trading on prices. It raises the questions of whether perfect competition is an adequate approximation when traders have private information. Here we investigate the strategic foundations for rational expectations equilibrium. We ask whether the Bayesian Nash equilibrium of a model with a finite number of traders converges to rational expectations equilibrium as the market becomes large. We study a one-period model in which traders submit demand schedules to buy or sell shares of a risky asset. Traders have two private signals, one about the asset’s value and the other about their endowments. Random endowments motivate risk-averse traders to trade for hedging motives. A large market is typically modeled as consisting of infinitely many traders with different private information. In this paper, we divide traders into groups so that each trader competes not only with the other groups, who have different signals, but also with his own group members, who who share the same signals. We assume random variables are normally distributed, and traders have exponential utility with the same risk aversion, endowment shocks with the same variance, and private information with the same precision. These symmetry assumptions make the model analytically tractable. Each group’s aggregative risk aversion, each group’s aggregate endowment shock, and the precision of each group’s information are held constant as the number of members per group increases. We find that a strategic equilibrium does not approach a rational expectations equilibrium as the number of groups or the number of competitors per group goes to infinity. Perfect competition and fully revealing prices cannot be achieved simultaneously. The economic intuition for this result is developed in the following six steps. First, we measure competition by the quantity a trader trades as a fraction of the quantity he would trade as a price taker, and we measure price informativeness by the fraction of other traders’ information a trader extracts from prices. In characterizing equilibrium, only three parameters matter: the number of groups, the number of competitors per group, and our measure of adverse selection, the ratio of private information to endowment shocks. 1

Second, price informativeness is solely a function of adverse selection; it does not depend on the number of groups or number of competitors per group. As infinitely many competitors share the same information, the market becomes perfectly competitive. We show that this limit corresponds to the model of Diamond and Verrecchia (1981). In equilibrium, the price remains partially revealing. Intuitively, more competition makes traders trade more aggressively on their private information. This does not affect price informativeness because traders trade equally more aggressively on their endowment shocks. Third, the market does not become perfectly competitive as infinitely many groups compete with different information. Since each group has unique private information, traders remain “large” because new information continues to move the price even when the group becomes “small” relative to the market. Perfect competition requires, but is not guaranteed by, infinitely many competitors sharing the same information. Fourth, as adverse selection increases, the price becomes more informative but the market becomes less competitive. The price becomes informative through the process by which each trader moves the price towards his own valuation. The more the price incorporates private information of a trader, the smaller the quantity he trades compared with a price taker. Thus, importantly, optimal exercise of market power has opposite implications for price informativeness and competition. Fifth, the existence of equilibrium requires adverse selection to be sufficiently small. Following the spirit of trembling hand perfect equilibrium, we add a vanishingly small amount of exogenous noise trading to a setup in which endowment shocks alone are too small for equilibrium to exist. This guarantees that there is always a well-defined equilibrium in the limit, even though the expected losses of noise traders are zero. In such an equilibrium, the market is infinitely noncompetitive in the sense that trade vanishes. Lastly, without endowment shocks, the price becomes fully revealing while the market remains infinitely noncompetitive as the number of competitors per group goes to infinity. With a finite number of traders, the price is partially revealing. Each trader incorporates a maximum of one half of his private information into prices. As the number of competitors increases to infinity, each trader continues to move the price and the price becomes fully revealing. Even when they become “small” in terms of private information, they remain “large” in terms of their effect on the price. Importantly, the price becomes fully revealing because the market remains noncompetitive. Almost2

perfect competitors trade large quantities like price takers, and they choose to do so because little of their private information gets incorporated into prices. Combining the six results, we confirm that either traders become price takers or the price becomes fully revealing, but not both, as the number of competitors who share the same information goes to infinity. The key intuition is the opposite implications for competition and price informativeness that result from strategic traders’ optimal exercise of market power. Such an interaction between price informativeness and competition is absent in the model of Grossman and Stiglitz (1980), who assume price-taking. The paradox in their model is that no trader has an incentive to acquire costly information because the price becomes fully revealing as noise trading vanishes. In our model, the price is always partially revealing with a finite number of traders; the paradox disappears. Our no-trade and noisy-price result with vanishing noise trading is different from the framework of Milgrom and Stokey (1982), who always find a fully revealing price with no trade. Moreover, even though we allow initial allocations to be Pareto inefficient, there may still be no trade despite substantial gains from trade. Whether there are better trading mechanisms for internalizing gains from trade is left for future study. Developing the strategic foundations for rational expectations equilibrium has been the topic of many papers. Wilson (1977), Milgrom (1981), Pesendorfer and Swinkels (1997), and Kremer (2002) study whether the price becomes fully revealing as the market becomes large in a model where buyers are strategic but sellers are not. Kyle (1989) assumes that informed traders are strategic, but trading is sustained by exogenous noise traders rather than endowment shocks. In a model in which traders can buy or sell one unit of an indivisible good, Reny and Perry (2006) show that rational expectations equilibrium is obtained as the market becomes large. Vives (2011) examines a model of strategic supply-function competition that is similar to our demand-schedule competition. Our paper is different from the closely related literature in three ways. First, we use exponential utility while Vives (2011), Rostek and Weretka (2012, 2015) and Bergemann, Heumann and Morris (2015) use quadratic storage costs. This is qualitatively different from exponential utility. With exponential utility, but not with quadratic storage costs, a more informative price makes the market less competitive by decreasing the riskiness of the asset. Second, in our model, each trader has two signals, and this prevents traders from inferring the average of other traders signals perfectly. In a model 3

of correlated private values, Vives (2011) assumes each trader has one signal about his own private value; symmetry makes the price fully revealing. Rostek and Weretka (2012, 2015) extend this approach to allow heterogeneous correlations among traders’ private values. Bergemann, Heumann and Morris (2015) study a more general informational environment that allows confounding between common and private components of a signal. They note such confounding is an important element in the informational environment. In our model, there is confounding between the two signals. Lastly, by dividing traders into groups, we clearly show how the effect of the number of groups differs from that of the number of competitors per group. The plan for this paper is as follows. Section 1 describes the setup of the model and defines an equilibrium. Section 2 characterizes an equilibrium and provides the conditions for existence and uniqueness of an equilibrium. Section 3 analyzes comparative statics when there is no noise trading and shows that price informativeness is independent of the number of competitors who share the same private information. Section 4 introduces an equilibrium with vanishing noise trading and shows that the market is always infinitely noncompetitive in such an equilibrium. Section 5 concludes.

1 Setup There is one round of trading in which traders exchange a single risky asset against a safe asset whose return is normalized to one. The exogenous liquidation value of the risky asset v is distributed N 0, σV2 with σV2 > 0. There are N symmetric groups of traders. Each group consists of M identical informed speculators. Altogether, there are M N traders indexed (m, n ). Each informed trader has exponential utility with constant

risk aversion parameter M ρ; thus, ρ > 0 measures the aggregate absolute risk aversion of each group. Before trading, each trader (m, n ) in group n obtains two identical signals. First, all traders in group n obtain an identical private signal about v given by 1/2

in = τI

v

σV

!

+ e n,

where

e n ∼ N (0, 1) .

(1)

Each signal provides imperfect information about the liquidation value v . Since var {in } = 1 + τI , the precision parameter τI is a signal-to-noise ratio in which the signal τI1/2 σvV has variance τI and the noise e n has variance equal to one. The error terms in the sig4

nals e 1, . . ., e N are distributed independently. This implies that private information is different across different groups n = 1, . . ., N even though it is the same for the M members of group N . Second, each trader in group n receives an identical random endowment of sm, n ≔ 1 s shares of the risky asset. The quantity sn is distributed independently across groups M n n with sn satisfying

sn ∼ N s¯n , σS2 ,

where

N X

s¯n = 0.

(2)

n =1

Each trader observes the realization of his own endowment sn , knows his own mean s¯n , and knows that the aggregate mean endowment is zero. Each trader infers the identical endowment of other members of his own group n but does not observe the endowment of any other group. The model is set up so that the group’s aggregate absolute risk aversion ( ρ), aggregate endowment shock (σS2 ), and quality of private information (τI ) do not change when the number of traders within each group (M ) changes. As M increases, more and more traders with the same private information compete with one another. At the same time, each trader becomes smaller in the sense that his risk bearing capacity ((M ρ)−1 ) is less. Varying the parameter M changes in the competitiveness with which trading on private information about payoffs and endowments takes place without changing other aspects of the economy. We refer to M as the number of competitors. Noise traders demand a random quantity z which is distributed N 0, σZ2 . Noise traders do not optimize anything; their trading is exogenous. Since the model does not require exogenous noise traders, the model allows σZ2 = 0. In fact, most of our analysis assumes that σZ2 is exactly zero or approaches zero, in which case the expected loss of noise traders is zero. Assume all random variables are jointly normally distributed so that v ; e 1, . . ., e N ; s 1, . . ., sN ; and z are all independently distributed. Except for different mean endowments s¯n , the model is symmetric in that it looks the same from the perspective of every informed trader. Before the realization of private information and endowment shocks, all traders have identical risk aversion, identical beliefs about the signal precision, and identical beliefs about the distribution of endowment shocks about their means. If we were to think of M as representing a continuum of perfect competitors, our model without noise trading would collapse to the model of Diamond and Verrecchia (1981). 5

Trading. After observing his own endowment shocks sn and private signal in , each trader (m, n ) submits a demand schedule X m, n (p | in , sn ). This notation means that X m, n is a function of the price p , and the function is measurable with respect to sn and in .

Let X denote the M × N matrix of submitted demand functions whose (m, n )th

element corresponds to X m, n . An auctioneer aggregates all M N functions to calculate a market clearing price, denoted p (X ), which satisfies the market clearing condition N X M X

X m, n (p ) + z = 0.

(3)

n =1 m =1

If there is no market clearing price, then there is no trade (xm, n = 0 for all (m, n )). If there are many market clearing prices, then the auctioneer chooses the smallest price which minimizes trading volume. Given the matrix of submitted demand schedules, trader (m, n ) realizes wealth w m, n (X ) := v

sn

+ (v − p (X )) X m, n (p (X ))

M

(4)

and achieves expected utility

um, n (X ) := E − exp (−M ρw m, n (X )) .

(5)

The model is described by seven exogenous parameters: M , N , τI , ρ, σV , σZ and σS . There are two dimensions of measurement: dollars and shares. The parameters σZ and σS have dimensions of shares, the parameter σV has dimensions of dollars-per-share, and ρ has dimensions of per-dollar. In what follows, we use ρ and σV as units to scale variables in dollars-per-share by σV and variables in shares by ( ρσV )−1 . Therefore it is crucial to assume ρ > 0 and σV > 0. The equilibrium concept is a Bayesian Nash equilibrium. An equilibrium is a matrix of demand schedules X such that (1) a market clearing price p (X ) is always well defined, and (2) for all m = 1, . . ., M and n = 1, . . ., N , trader (m, n ) chooses his demand schedule X m, n to maximize his expected utility um, n (X ), taking as given the demand schedules of the other M N − 1 traders. Define a symmetric linear equilibrium as an equilibrium in which all traders choose

6

the same linear demand schedule

ρσV X m, n (p | in , sn ) = πC − πS ρσV sn + πI in − πP

p

σV

,

(6)

where the four endogenous parameters πC , πS , πI , and πP define the same linear function X m, n for all m = 1, . . ., M and n = 1, . . ., N . Multiplying X m, n (p | in , sn ) and sn by ρσV and dividing p by σV scales the four parameters πC , πS , πI , and πP to make them dimensionless. This scaling convention is used throughout this paper. If M N πP = 0, then every trader submits a totally inelastic demand schedule, and the resulting aggregate demand is either identically zero or some random quantity which is non-zero with probability one. Market clearing requires this aggregate demand to be identically zero; this further requires noise trading to be zero with probability one (σZ2 = 0) and each trader’s demand to be identically zero (X m, n ≡ 0, for all m, n ). In such a no-trade equilibrium, the market clearing price is not uniquely determined since any price can support the allocation. Such an equilibrium exists whenever σZ2 = 0. We call this a trivial no-trade equilibrium and exclude it from the following analysis. Our goal is to characterize existence and uniqueness of symmetric linear equilibria. Discussing asymmetric equilibria or equilibria with non-linear strategies takes us beyond the scope of this paper.

2 Equilibrium To understand equilibrium, it is intuitively necessary to model how learning from prices and exercising market power simultaneously affect the demand schedules traders choose. To accomplish this, we introduce a “price informativeness” parameter and a “market noncompetitiveness” parameter and then discuss how the values of these two parameters trade off against each other. This discussion leads to the counterintuitive result that a more informative price goes together with a less competitive market. The equilibrium solution proceeds in five steps using the no-regret pricing approach. A trader (1) observes his residual supply schedule, (2) learns about other traders’ private information from the intercept of this schedule, (3) finds the optimal quantity on his residual supply schedule, (4) and implements this optimal quantity by submitting a demand schedule, which (5) is the same as the demand schedules conjectured for 7

other traders. Trader (m, n ) conjectures and takes as given symmetric

Residual Supply Schedule.

linear demand schedules for the other traders, described by the four endogenous parameters πC , πI , πS and πP as in (6). Ruling out trivial no-trade equilibria as discussed above (M N πP , 0), the market clearing condition (3) implies that trader (m, n ) has a well-defined residual supply schedule given by p

σV

=

p m, n

σV

+

1 ρσV xm, n . (M N − 1) πP

(7)

The intercept p m, n is given by p m, n

σV

=

P

(m ′, n ′),(m, n ) (πC

+ πI in ′ − πS ρσV sn ′ ) + ρσV z . (M N − 1) πP

(8)

Trader (m, n ), by submitting a demand schedule, is able to condition the quantity demanded on the market clearing price. This implies that the trader can choose the optimal demand as if he already observes the market clearing price p as well as the price that would prevail if he did not trade p m, n . In units of dollars per share, (7) can be written p = p m, n + λ x m, n ,

with

λ=

ρσV2 , (M N − 1)πP

(9)

where λ is the price impact parameter defined in Kyle (1985) and Kyle (1989). Price impact λ measures how much the per-unit price of the risky asset changes in response to the informed trader’s buying one more share. Since the model is symmetric about zero, this implies that the constant πC in the demand schedule can be shown to be zero in equilibrium, regardless of the values of other parameters. For the purpose of exposition, we assume πC = 0 without loss of generality. Learning from the Price. Trader (m, n ) infers other traders’ private information from the intercept p m, n in (8). Let τ∗ denote the dimensionless ratio of the prior variance

8

(σV2 ) to the posterior variance (var v p m, n, in , sn ) of the liquidation value given by ∗

τ =

σV2

. var v p m, n , in , sn

(10)

Symmetry implies that τ∗ is the same across all traders. Since the posterior variance is at least as accurate as the prior variance, the inequality τ∗ ≥ 1 holds by definition. Trader (m, n )’s learning from the price is described by the following lemma. All proofs are in the Appendix. Lemma 1 (Learning From Prices.). Assume (N − 1) τI πI , 0. Then τ∗ can be written

τ∗ = 1 + τI + (N − 1) τI ϕ,

(11)

where ϕ is given by

πS ρσV σS ϕ = *1 + πI ,

!2

1 ρσV σZ + N −1 M πI

!2

+ -

−1

.

(12)

Trader (m, n )’s conditional expectation of the fundamental value is given by E

(

v

p m, n , i n , s n σV

)

√

τI (N − 1) ϕπP p m, n M − 1 πP p m, n πS = ∗ in + − ϕ in − − ρσV sn τ πI σV M πI σV πI (13)

The dimensionless endogenous parameter ϕ given by (12) is both intuitively and analytically important in this paper. From (10), the parameter ϕ ∈ [0, 1] intuitively measures how much information about v trader (m, n ) extracts from prices as a fraction of the total precision of other traders signals; we refer to ϕ as “price informativeness.” If ϕ = 0, then no information is extracted. If ϕ = 1, then the maximum amount of information is extracted. When traders are identical (N = 1), traders have no private information (τI = 0), or traders do not trade on the private information they have (πI = 0), then there is no learning from the price. We set ϕ = 0 in these cases. When there is more than one trader in each group (M > 1), the intercept of trader (m, n )’s residual supply schedule p m, n is already a function of his own private information in and endowment shock sn because of the demand schedules submitted by the other M − 1 traders in the group. In the valuation of the risky asset (13), trader (m, n ) 9

!!

.

uses in and sn to eliminate the effect of trading by his own group members to extract the private information of other groups accurately. Optimal Quantity Traded. All random variables are jointly normally distributed, and trading strategies are linear. Thus, the trading strategy which maximizes exponential utility in (5) is the same as the trading strategy which solves the quadratic maximization problem Mρ max E w m, n (xm, n ) p m, n , sn, in − var w m, n (xm, n ) p m, n, sn, in . x m, n 2

(14)

Using (7) and (10), the first-order condition is E

(

v

)

p m, n , s n , i n − − ∗ ρσV sm, n σV σV τ p m, n

M

! M 2 − + ρσV xm, n = 0. (15) (M N − 1) πP τ∗

Define the “market noncompetitiveness” parameter χ by1

χ :=

τ∗ τ∗ λ. = (M N − 1) πP M M ρσV2

(16)

Then the second-order condition can be written as 1 χ>− . 2

(17)

If χ < −1/2 holds, the trader can obtain infinite utility by buying or selling an infinite quantity, and hence an equilibrium does not exist. To provide the intuition for χ, define a trader’s “target inventory” by TI sm ,n ≔

E v p m, n , i n , s n − p m, n M ρσV2 /τ ∗

.

(18)

TI The target inventory sm , n is the quantity a trader would demand to hold if he traded like a perfect competitor, ignoring price impact resulting from a nonzero slope λ of

1The second inequality in (16) gives an alternative way of writing χ using the definition of λ from (9). In the one-period model of Kyle (1985), the informed trader is a monopolist. This corresponds to χ → ∞, with the informed trader moving the price to a point half-way between his value and his expected price if he does not trade. In Kyle (1989), the “information incidence” parameter ζ measures the change in the equilibrium price as a response to the change in the trader’s valuation. When M = 1, this is similar to parameter χ/(1 + 2 χ), which is always positive but less than one-half.

10

the residual demand schedule. If the trader achieves the target inventory, he would choose not to trade. Substituting (18) into the first-order condition (15) yields the optimal quantity traded 1 TI x m, n = sm − s . (19) m , n ,n 1 + 2χ The fraction 1/(1 + 2 χ) measures the optimal quantity a trader chooses to trade as a fraction of how much he would trade if he were a price taker. If χ = 0, the market is perfectly competitive; if χ → ∞, , the market is infinitely noncompetitive, and traders

trade a vanishingly small fraction of the quantity needed to achieve their target inventories. We show in Theorem 1 that χ is nonnegative in equilibrium.2

The endogenous parameter χ is a scaled ratio of market impact to risk aversion. When χ is small, risk aversion restricts trading more than market impact, and thus traders trade quantities closer to target levels. When χ is large, market impact restricts trading more than risk aversion, and traders trade only a small fraction of the quantity needed to reach target inventories. Market noncompetitiveness χ also determines the extent to which a trader moves the price towards his valuation of the risky asset: p − p m, n

1 1 1− = 2 1 + 2χ

!

E v p m, n , i n , s n −

M ρσV2

τ∗

!

s m, n − p m, n .

(20)

Trader (m, n ) optimally chooses how much to move the price from p m, n , the prevailing price if trader (m, n ) did not trade, towards his own valuation of the risky asset ad justed for his endowment shock (E v p m, n , in, sn − (M ρσV2 /τ∗ )sm, n ). If the market is very competitive ( χ close to zero), the trader does not move the price much (p close to p m, n ). When the trader exercises great monopoly power ( χ close to infinity), a trader

will move the the price more, to a level almost halfway between the prevailing price and his valuation of the asset. 1 1 1 Comparing the quantity factor 1+2 from (20) with the price factor 1 − χ 2 1+2 χ from (19) shows that market noncompetitiveness χ has opposite effects on quantity and price. When noncompetitiveness χ increases, a trader chooses to restrict his trad2This paper supersedes an earlier version Kyle and Lee (2016) which considers a more general model in which traders have heterogeneous prior beliefs on the precision of their signals. In the special case when all traders are relatively underconfident and M ≥ 2, χ can satisfy − 12 < χ <, 0 in equilibrium. This paper’s common prior setup implies χ ≥ 0 in equilibrium. Kyle, Obizhaeva and Wang (2016) consider a model with disagreements but no endowment shock and no noise trading.

11

ing more, but the price incorporates a larger fraction of his information. A trader chooses to trade a competitive quantity only when his private information does not get incorporated into the price. As an index of noncompetitiveness, χ may be broadly applied outside our symmetric model to compare how much illiquidity affects the trading of different traders in different markets. To evaluate and refine trading strategies, asset managers often calculate the ratio of market impact costs to expected trading profits (alpha). The parameter χ measures exactly this ratio. It is a trader’s dollar cost of reaching target inventory as a fraction of a “paper-trading” profit (dollar profit in the absence of price impact). To apply χ to asset management, think of the reciprocal of risk aversion as measuring assets under management. Market noncompetitiveness χ is high when assets under management are high, market impact is high, or learning significantly increases the precision.3 This effect of learning on market noncompetitiveness is absent in the quadratic storage cost model used by Vives (2011), Rostek and Weretka (2012, 2015) and Bergemann, Heumann and Morris (2015). We can define a hypothetical quadratic storage cost parameter γ by γ := ρσV2 /τ∗ . If τ∗ were a constant, then the quadratic storage cost model and the exponential utility model would be the same. In fact, τ∗ defined by (10) increases in exogenous parameters τI and N and endogeneous parameter ϕ. Therefore, when the amount of learning varies due to changes in τI , N , or ϕ, the two models obtain different results. According to (16), if τ∗ were a constant, then χ would be simply proportional to λ . In this sense, with quadratic storage costs, a more informative price does not directly reduce competition. Trader (m, n ) can implement his optimal quantity xm, n by substituting his valuation of the risky asset (13) into the optimal quantity traded (19) to obtain the best response strategy X m, n to the symmetric strategies of other traders. Lemma 2 (Best Response). Assume χ > − 12 . If the other M N − 1 traders chose strategies defined by the parameters πI , πP , and πS , then trader (m, n )’s best response is the optimal 3Risk tolerance 1/ (M ρ) intuitively corresponds to assets under management. For a small mean µ and small variance σ 2 , the competitive demand function for a log-utility investor with wealth W is approximately W µ /σ 2 . When this is compared to the CARA-normal competitive demand ρ−1 µ /σ 2 , it is easy to see that 1/ ρ maps directly into wealth W .

12

demand schedule X m, n (p | in , sn ) given by (N − 1) ϕτI1/2 πP p ϕτ1/2 *1 + χ + I + M ρσV X m, n (p | in , sn ) = τ1/2 in − *τ∗ − + − ρσV sn I π M 2 πI I , , - σV ! (M − 1) ϕτI1/2 πP p πS − − ρσV sn , in − M πI σV πI (21) where τ∗ , ϕ and χ are given by (11), (12) and (16) respectively. Characterization of Equilibrium. A linear symmetric equilibrium is found by equating a trader’s best response (21) to the strategy the trader conjectures that others are playing. An equilibrium can be fully characterized using the exogenous parameters and the single endogenous informativeness parameter ϕ. Theorem 1 (Characterization of Symmetric Linear Equilibrium). Suppose ρ > 0, σV > 0, and M N > 2. If (N − 1) τI > 0, then the set of symmetric linear equilibria, excluding trivial no-trade equilibria, is characterized by the set of all endogenous parameters ϕ such that (1) ϕ solves 1 − ϕ ( ρσV σS )2 ( ρσV σZ )2 / (N − 1) = + 2 , N −2 M N +N −2 ϕ τI τI M − ϕ M N −1 M N −1

(22)

and (2) ϕ satisfies the second-order condition

ϕ<

MN − 2

MN + N − 2

(23)

.

If (N − 1) τI = 0, an equilibrium is characterized by ϕ = 0.

With τ∗ given by (11), the equilibrium demand schedule of trader (m, n ) is given by

! (M N − 2 − (M N + N − 2) ϕ) 1/2 τ∗ p ρσV X m, n (p | in , sn ) = τI in − ρσV sn − , M (M N − 1) 1 + (N − 1) ϕ σV (24) the market clearing price is given by ! N N X 1 + (N − 1) ϕ * 1/2 X = τI in − ρσV sn + σV N τ∗ i =1 n =1 , p

13

M N −2 M N −1

ρσV z +, − MMN N+N−1−2 ϕ -

(25)

and market noncompetitiveness χ is given by

χ=

1 + (N − 1) ϕ . M N − 2 − (M N + N − 2) ϕ

(26)

Equation (22) defines the endogenous parameter ϕ as the solution to an equation whose other parameters are all exogenous. Once ϕ is determined, χ is expressed as a simple function of ϕ, M and N in (26). Price informativeness ( ϕ) and market noncompetitiveness ( χ) together capture the important aspects equilibrium quantities and prices. We discuss how exogenous parameters determine equilibrium ϕ and χ in detail in the next section. In (22), there are two dimensionless products of dimensional quantities: ρσV σS and ρσV σZ . Changing units of measurement has no real effect on (22), (24), (25), and (26). This implies ρ and σV can be interpreted as scaling variable with ρ = σV = 1 assumed without loss of generality. Thus, if the four dimensional exogenous variables ρ,

σV , σS , and σZ , change in such a way that the two dimensionless products ρσV σS and ρσV σZ do not change, then ϕ does not change, and the properties of the equilibrium do not change in many respects.4 Existence and Uniqueness. Theorem 1 can be used to characterize existence and uniqueness in terms of exogenous parameters only. Corollary 1 (Existence and Uniqueness). Assume ρ > 0 and σV > 0. Then there exists a symmetric linear equilibrium, excluding trivial no-trade equilibria, if and only if M N > 2 and at least one of the following three conditions holds: (N − 1) τI = 0,

(27)

σZ2 > 0,

(28)

τI 2

( ρσV σS )

2 N

.

(29)

If a symmetric linear equilibrium exists, it is unique. 4This property is shared by many finance models. Fundamental model properties depend on the ratio of the risks to be borne—measured by σV σZ and σV σS —to dollar risk-bearing capacity ρ−1 . For example, ρσV σZ or ρσV σS can become small either because risk bearing capacity increases ( ρ becomes small) or because the risks to be borne σV σZ and σV σS become small. Either way, the effect on equilibrium is the same.

14

A symmetric linear equilibrium exists when either (1) there is no information asymmetry, in which case ϕ = 0 is defined by continuity (in condition (27)); or (2) there is any amount of noise trading (in condition (28)); or (3) endowment shocks are large enough, relative to the precision of private signals, to prevent the market from shutting down due to too much adverse selection (in condition (29)). These conditions are equivalent to the second-order condition (23), which in turn results from (1) maintaining concavity of the objective function in (14) and (2) ruling out trivial no-trade equilibria (M N πP , 0). Uniqueness follows from (22) as its left hand side is strictly decreasing in ϕ and the right hand side is weakly increasing in ϕ. While the endowment shock needs to be sufficiently large for an equilibrium to exist, an arbitrarily small amount of exogenous noise trading guarantees existence of equilibrium. Unlike optimizing traders who receive endowment shocks, noise traders are willing to incur whatever trading costs are necessary to sustain an equilibrium with trade.

3 Competition and Price Informativeness In this section we discuss how price informativeness and competition depend on exogenous parameters when there is some information asymmetry ((N − 1) τI > 0) but no noise trading (σZ = 0). While there remain six exogenous parameters (M , N , ρ, σV , σS and τI ), equations (22), (23), and (26) show that characterizing ϕ and χ only requires the three exogenous parameters M , N , and “adverse selection” θ , defined by

θ :=

τI ( ρσV σS )2

.

(30)

This parameter θ ∈ (0, ∞] is the ratio of private information to endowment shocks. A higher precision of private information (τI ) increases adverse selection; a higher variance of endowment shocks (σS2 ) reduces adverse selection.

Corollary 2. Assume (N − 1) τI > 0 and σZ2 = 0. Then a symmetric linear equilibrium exists if and only if the second order condition

θ

15

2 N

(31)

holds. If equilibrium exists, price informativeness ϕ is given by

ϕ=

θ , 1+θ

(32)

and market noncompetitiveness χ is given by

χ=

1+ Nθ . MN − 2 − Nθ

(33)

If equilibrium exists for particular values of M , N and θ , then it exists for larger M , larger N , and smaller θ . We discuss next the comparative statics results for the three exogenous parameters M , N and θ . The Number of Competitors (M ).

Increasing the number of competitors M implies

increasing competition. Not surprisingly, (33) implies that market noncompetitiveness χ monotonically decreases as M increases. As M → ∞, equilibrium outcomes approach perfect competition since χ → 0, and the market becomes perfectly com-

petitive in the sense that all traders, acting as price takers, achieve their target inventories in one round of trading (see (19)); this equilibrium corresponds to the model of Diamond and Verrecchia (1981). The number of competitors (M ) indeed captures the notion of competitiveness of the market. More surprisingly, (32) implies that price informativeness ϕ is not affected when the number of competitors M changes. Price informativeness ϕ is determined solely by adverse selection θ . It does not depend on how many traders share the same private information or how many different groups have different private information. This is counter to the conventional wisdom that more competition makes prices more informative by inducing traders to shade their bids less and thus trade more aggressively. While traders trade more aggressively on private information as M increases, this does not lead to more informative price because traders also trade equally more aggressively on their endowment shocks. The proportion by which traders shade their trading, as a function of M , is the same for private information and endowment shocks because optimal exercise of market power is governed by the same incentives for both private information and endowment shocks. The Number of Groups (N ). Setting the model up with N groups of M traders distinguishes the effects of changing the number of groups from changing the number of 16

members of each group. Like increasing the number of competitors M , (33) implies that noncompetitiveness χ decreases as the number of groups N increases. Unlike the result for the number of competitors M , as the number of groups with different private information (N ) goes to infinity, equilibrium outcomes do not approach perfect competition. Taking the limit as N approaches infinity in (33) yields

χ→

θ >0 M −θ

as

N → ∞.

(34)

The economic intuition for why the market remains imperfectly competitive, even when the total number of traders goes to infinity, is that each group’s uniquely different private information is known to only a finite number of traders M . Traders maintain incentives to trade less aggressively because they have market power over their private information. With M = 1, the equilibrium is like monopolistic competition. Like the corresponding result for changing M , changing N does not change price informativeness ϕ. Adverse Selection (θ ). Adverse selection (θ ) affects both price informativeness and competition. Equation (32) implies that price informativeness ( ϕ) depends only on θ and increases in θ . More adverse selection increases the informativeness of prices. Equation (33) implies that market noncompetitiveness ( χ) depends on M , N and θ . When M and N are fixed and θ varies, χ increases in θ , and thus increases in ϕ. This implies an inverse relationship between price informativeness and competition: As adverse selection θ increases, a more informative price (larger ϕ) is associated with less competition (larger χ). This inverse relationship between price informativeness and competition may seem counterintuitive. Imperfect competition is mainly driven by information asymmetry. As the price becomes more informative, each trader knows more about others’ information, and there is less information asymmetry. One might think that therefore the market should be more competitive. Our earlier discussion on the opposing effects of competition on the price and the quantity, as shown in (19) and (20), provides economic intuition for this result. We can also examine the positive relationship between ϕ and χ in the best response strategy (21). Suppose all traders have the same conjectures about other traders’ strate-

17

gies, given by πI , πP , πS and the resulting ϕ and χ defined by (12) and (16). By aggregating across traders, we find the best-response value of market noncompetitiveness, denoted BR ( χ). Since traders share the same conjectures, the resulting best response strategies are the same as well. To reduce the dimensionality of the best responses, we impose a restriction that the ratio πP /πI is a fixed point conditional on ϕ. The assumption χ ≥ 0 ensures the existence of a best response. Corollary 3. Suppose each trader (m, n ), for all m and n , takes as given other players’ strategies πI , πP and πS that satisfy

πP 1 + τI + (N − 1) τI ϕ = πI (1 + (N − 1) ϕ) τI1/2

and

χ ≥ 0,

(35)

where ϕ and χ are given as (12) and (16), respectively. Then the market noncompetitiveness that results from the best response strategy is given by 1 + τI + (N − 1) τI BR (ϕ) BR ( χ ) = 1 + τI + (N − 1) τI ϕ

!

! 1+ χ ϕ (1 + χ) ϕχ + + . M N − 1 M − (M − 1) ϕ M − (M − 1) ϕ (36)

The noncompetitiveness that results from the best response strategies is given by the product of the two factors of the right-hand side in (36). The first factor is a coefficient that is independent of χ. This coefficient is greater than one if the price informativeness that results from the best response strategies (BR ( ϕ)) is higher than the conjectured price informativeness ( ϕ). If the conjectured price informativeness is a fixed point ( ϕ = BR ( ϕ)), which is the case when the conjectured strategies are already in equilibrium, the coefficient reduces to one. There are three terms in the second factor on the right-hand side of (36). The first term is the pure effect of competition, which corresponds to the case when there is no private information ((N − 1) τI = 0). When all traders conjecture χ = 0, the best response strategy yields M N1 −1 because all traders are risk averse and owning the asset is

risky. As the conjectured χ increases, the best response χ increases. The condition that there are strictly more than two traders (M N > 2) guarantees that the slope is strictly less than one. The second and the third terms reflect the effect of price informativeness on competition. The distinction between the two terms comes from the fact that, in a double auction, all traders are demanders as well as suppliers of liquidity. As a liquidity supplier, a trader is concerned about suffering the winner’s curse. When other traders 18

want to buy an asset from the trader, it is likely that the valuation of the asset is higher because other traders may have different information from him. As a liquidity demander, the trader needs to compensate other traders for the winner’s curse that they may suffer as a result of his own private information. This compensation disappears if the conjectured χ is zero because trading becomes costless. The two winner’s curses—as a liquidity supplier and as a liquidity demander—become more severe as the price becomes more informative. This feeds back into the best response strategy and results in the inverse relationship between price informativeness and competition. Collecting all terms, (36) shows how and why price informativeness and competition are inversely related in equilibrium. This also explains the equilibrium existence condition. When price informativeness is a fixed point ( ϕ = BR (ϕ)), the slope of χ is the sum of all three coefficients in the second line. The sum needs to be strictly less than one for an equilibrium to exist. This is equivalent to the equilibrium second-order condition (23).

4 Equilibrium with Vanishing Noise Trading When there is no noise trading (σZ = 0), equilibrium does not exist unless adverse selection θ is sufficiently small (θ < M − N2 ). This nonexistence result suggests that some issue needs to be addressed in our model. While informal intuition may suggest that there is no trade when equilibrium fails to exist—and indeed, a trivial no-trade equilibrium does exist is this case—nonexistence does not formally imply no trade. Instead, nonexistence represents failure of the model to make a prediction even though no trade is a possible equilibrium outcome. We address this modeling issue by defining an equilibrium without noise trading to be the result of taking a limit as noise trading vanishes (σZ → 0). We interpret van-

ishing noise as adding small perturbations to the trading environment. While this is in the spirit of trembling hand perfect equilibrium, it involves perturbations of the players’ actions rather than the exogenous perturbations which we propose. According to Corollary 1, any arbitrarily small amount of exogenous noise trading allows an equilibrium to exist (because (28) is satisfied), despite the presence of an arbitrarily large amount of adverse selection (which makes (29) fail to be satisfied). Confirming intuition, there is indeed no trade in such a limit. Contrary to intuition, the price remains noisy even in the limit when both endowment shocks and noise trading vanish. This 19

result contrasts with the fully revealing prices obtained by Milgrom and Stokey (1982) and the price taking assumed by Grossman and Stiglitz (1980). The next theorem formalizes this result. Theorem 2. Let E denote an economy described by {M , N , θ, σZ } such that M N > 2,

(N − 1) τI > 0, and σZ = 0. Let (Ek )k∞=0 = (E0, E1, . . . ) denote an arbitrary sequence of economies such that Ek has exactly the same exogenous parameters as E for all k except that σZ2 (k ) > 0 and σZ2 (k ) → 0 as k → ∞. Then there exists a unique equilibrium in Ek for all k . Moreover, price informativeness ϕk in economy Ek satisfies (

θ MN − 2 ϕk → ϕ := min , 1 + θ MN − 2 + N

)

as

k → ∞,

(37)

)

as

k → ∞.

(38)

and market noncompetitiveness χk satisfies −1

χk → χ

−1

:= max

(

MN − 2 − Nθ

1 + Nθ

, 0

Comparing Theorem 2 with Corollary 2 shows that adding vanishing noise trading does not have any effect when an equilibrium already exists without vanishing noise (θ < M − N2 ). When an equilibrium does not exist without noise trading, adding vanishing noise trading keeps the price sufficiently noisy to support an equilibrium that is infinitely noncompetitive. The price is noisy enough so that the second-order condition is satisfied for all k . This result is unique to noise trading; a similar result is not obtained from vanishing endowment shocks. Although the limit as noise trading vanishes has no trade, this limit is not a trivial no-trade equilibrium because the noisy price is well-defined in the limit. Noisy Price With Vanishing Noise. When an equilibrium does not exist without vanishing noise (θ ≥ M − N2 ), price informativeness is independent of adverse selection θ

and is determined by M and N . The economic intuition for this result is that in such an equilibrium, the market becomes infinitely noncompetitive (since χ → ∞ from (38)). This implies that each trader incorporates half of his private information into prices 1 1 (since 12 1 − 1+2 χ → 2 in (20)). If all traders with the same private information collude (equivalent to M = 1), this

implies that price informativeness ϕ satisfies ϕ ≈ 1/2 for large enough N (or, more precisely, ϕ = 12 1 − N 1−1 ). This corresponds to the monopolistic competition limit 20

in Kyle (1989) as the number of different informed traders (N ) approaches infinity. As the number of non-colluding, oligopolistic competitors within each group increases to M = 2, 3,. . ., the informativeness of prices ϕ increases to approximately 2/3, 3/4, 4/5,. . ., for large enough N . While each trader still incorporates half of his marginal private information into prices, the price becomes more informative when the overlapping private information is shared among many traders M . The outcome of this oligopolistic competition among M competitors resembles quantity competition in a Cournot equilibrium in which each firm tries to maximize its profit by supplying only the half of its residual demand. As the number of firms increases in quantity Cournot competition, each firm becomes a smaller fraction of the market, and the total quantity produced increases to fractions 2/3, 3/4,. . ., of the quantity with perfect competition. Of course, the important distinction here is that even when traders become “small” in terms of their private information, they remain “large” in the market because their trading costs are constrained by vanishingly low market liquidity, not by risk aversion. Price Impact with Vanishing Noise. The equilibrium with vanishing noise replaces a trivial no-trade equilibrium—which has no well-defined price—with a non-trivial notrade equilibrium in which the price is well-defined as a limit. Mathematically, the difference between taking a limit and not taking a limit shows up in the denominator of the σZ -term in the ϕ-equation (22). When a limit is taken, both the numerator and the denominator of the σZ -term in (22) converge to zero, but the ratio has a well-defined limit given by ( ρσV σZ )2 / (N − 1) 1 → max 2 M − N −2 τI M − MMN N+N−1−2 ϕ M N −1

2 N

1 − , 0 θ

as

σZ → 0.

(39)

If θ > M −2/N , implying an equilibrium would not exist without noise trading, this limit is strictly positive. While noise trading vanishes, the price noise created by vanishing noise remains strictly positive. The intuition for this result is that the price impact of noise trading goes to infinity as σZ → 0.

According to (25), the price impact of noise trading λ Z —the per-share price change

21

in response to per-share noise trading—is defined by ! ρσV2 1 + (N − 1) ϕ * λZ ≔ +. M N −2 M N +N −2 N τ∗ − ϕM N −1 , M N −1

(40)

When θ ≥ M − 2/N , the price impact of vanishing noise trading goes to infinity (see (37).) The limit result (39) implies that even though the expected losses of noise traders E {z · λ Z z } vanish as noise trading vanishes, because

E { z · λ Z z } = λ Z σZ2 → 0

as

σZ → 0,

(41)

the variance of the price created by noise traders (var { λ Z z }) does not vanish, because (M N − 1)2 (N − 1) τI σV2 * 1 λ Z σZ → max 2 ∗ ,M − (M N − 2 + N ) τ 2

2

2 N

1 − + , 0 θ-

as

σZ → 0. (42)

Mathematically, this limit is finite because price impact λ goes to infinity at the same rate as σZ goes to zero. Since the price noise created by vanishing noise trading does not vanish, the price is sufficiently uninformative to allow an equilibrium to exist at the limit as noise trading vanishes. No-Trade Theorem.

In the vanishing-noise equilibrium, the market is infinitely non-

1 competitive ( χ → ∞), and thus there is no trade (since 1+2 χ → 0 in (19)). For σZ > 0, the second-order condition (23) holds as a strict inequality, but the second-order condition holds as an exact equality in the limit σZ → 0. Even though the price is well-

defined in the limit, the at-the-limit-strategies X m, n (p ) ≡ 0 themselves define a trivial no-trade equilibrium in which there is no well-defined price. Recall that initial endowments can have unequal, deterministic components sn in (2). Thus, it is common knowledge that there are ex-ante gains from trade when initial allocations are different. Even though there may be large potential gains from trade

to equalize inventories across traders, traders will not participate in any trade at all. When adverse selection (θ ) is sufficiently severe (i.e., θ ≥ M − N2 ), there is no trade even when there is a small endowment shock or when many traders have the same private information. The economic intuition for this market breakdown is that the market is so illiquid 22

that any attempt to trade away from inefficient endowments would move the price infinitely against the traders. While this is similar to the lemons problem described by Akerlof (1970), the nature of information asymmetry is different because Akerlof assumes that only the sellers have private information. We show that even when each trader has private information in a symmetric fashion, the market may fail to realize ex-ante gains from trade. Our no-trade theorem can be contrasted with other no-trade results which obtain fully revealing prices in the absence of noise trading. While Milgrom and Stokey (1982) allow more general preferences and distributions of random variables, they do not specify a mechanism for determining prices.5 Moreover, they assume the initial allocation of resources is Pareto optimal. In contrast, our model allows inefficient initial allocations and also requires the market-clearing equilibrium price to be defined by the well-defined mechanism of aggregating demand schedules. While we obtain no trade, the price in such an equilibrium cannot be fully revealing for any finite M . The price cannot be more informative than (37). The equilibrium in demand schedules is attractive because all traders are treated symmetrically and limit orders are protected. These are properties of well-functioning markets which organized exchanges and their regulators strive to implement. From the perspective of welfare economics, the main weakness of the equilibrium in demand schedules is that modest adverse selection can make trade break down even when there are large gains from trade due to large non-stochastic initial endowments. Whether there are better trading mechanisms for internalizing gains from trade is an interesting issue. Liu and Wang (2016) examine a dealer-market model in which dealers make profits by buying at the bid and selling at the offer while customers are not allowed to trade with one another at the the same price (such as the midpoint of the bid-ask spread). The monopolistic spread profits earned by dealers may allow trade to occur when it would not occur in our equilibrium in demand schedules. Duffie and Zhu (Forthcoming) study a workup process that allows traders to trade at fixed prices which do not necessarily clear the market. Glode and Opp (2016) study the welfare effects of trading with intermediation chains. Malamud and Rostek (2016) study the welfare effects of decentralized exchanges when traders have heterogeneous risk aversion. 5Tirole (1982) considers both static and dynamic settings. Dow, Madrigal and da Costa Werlang (1990) emphasize market completeness and common knowledge.

23

Comparative Statics: Large Number of Traders. Defining an equilibrium using a limit when noise trading vanishes makes it possible to describe comparative statics properties of the equilibrium when there is no noise trading and equilibrium would otherwise not exist because there is too much adverse selection (θ > M − N2 ). For finite adverse selection (θ < ∞), the effects of changing the number of traders

per group M or the number of groups N follow from Theorem 2. As M increases, the second order condition (θ < M − N2 ) becomes more relaxed and must eventually hold for sufficiently large M . Equation (38) implies that the equilibrium becomes competitive in the sense that χ → 0 as M → ∞. Equation (37) implies that when M is large enough for the second order condition to be satisfied, price informativeness ϕ is determined by θ alone and the price remains noisy in the sense that ϕ → θ /(1 + θ ) as M → ∞. As N increases, the second order condition (θ < M − N2 ) needs not be satisfied.

While increasing N does relax the second order condition, its effect is much more limited. If (and only if) θ is sufficiently large (θ ≥ M ), the market remains infinitely noncompetitive as N → ∞ (see (38).) The price becomes more informative as the number of groups increases but remains noisy with ϕ → M / (M + 1) as N → ∞ (see (37)). If endowment shocks are zero (σS = 0), then adverse selection is infinite (θ = ∞)

and the second order condition is never satisfied. For all M , equilibrium remains infinitely noncompetitive and there is no trade. In the limit M → ∞, the price becomes fully revealing as half of each trader’s private information is incorporated into the price. Fully revealing prices and price taking are mutually exclusive with strategic trading. This result is the opposite of competitive rational expectations models such as Grossman and Stiglitz (1980), Hellwig (1980), and Diamond and Verrecchia (1981), where in the limit as noise trading vanishes, (1) traders remain price takers and (2) prices become fully revealing. These limiting results might be justified with the following informal logic: Competitive rational expectations models are an approximation to what would happen in an oligopolistic model with an arbitrarily large number of traders. As noise trading vanishes, the noise from prices will disappear. Thus, it is reasonable to believe that price noise will disappear and price taking behavior will apply in a noncompetitive model with a large number of traders observing the same private information. Our model articulates a flaw in the preceding argument. In a model with private information, the competitiveness of the market cannot be assumed exogenously because 24

both market competitiveness and price informativeness are determined by strategic trading motives. The resulting inverse relationship between competition and price informativeness is so important that the market may remain noncompetitive even as infinitely many traders compete over the same information. Moreover, the model of Grossman and Stiglitz (1980) entails a well-known paradox that no trader has an incentive to acquire costly information because the price is fully revealing as noise trading vanishes. This paradox disappears in our model because the price is always partially revealing with a finite number of traders. Order of Limits. Our symmetric model is in general simpler than asymmetric models in which traders differ in the quality of their information. For example, Grossman and Stiglitz (1980) describe an asymmetric model whose equilibrium is more analytically complicated due to having both informed and uninformed traders. Our model with N = 2, σS2 = 0, and σZ2 > 0 resembles theirs as M → ∞, with the informed and uninformed replaced by two groups of symmetrically informed traders with independent private signals. The approach of competitve models such as Grossman and Stiglitz (1980), Hellwig (1980), and Diamond and Verrecchia (1981) implicitly takes the limit M → ∞ first. Taking the limit as M → ∞ in (22) and (23) for σZ > 0 yields (1 − ϕ)3 =

( ρσV σZ )2 ϕ, (N − 1) τI

where

ϕ<1.

(43)

This equilibrium corresponds to a symmetric version of the equilibrium of Hellwig (1980). Since ϕ < 1, (26) implies χ → 0 for any positive amount of noise trading σZ so that the market is perfectly competitive. Next, taking the limit σZ → 0 in (43) makes the price fully revealing since ϕ → 1. By first assuming M to be infinity, their analysis completely assumes away the delicate interaction between price informativeness and market competition. The fact that different results are obtained when limits are taken in different orders implies that different results are also possible when a double limit is taken with both

σZ → 0 and M → ∞. Suppose very small target values χ and 1 − ϕ, denoted χˆ and (N −1) τI (1−ϕˆ)3 1 2 1 − ϕˆ, are given. Now choose M = χˆ (1− ϕˆ) and σZ = ρ2 σ 2 ϕˆ . Then it can be shown V

from Theorem 1 ((22) and (26)) that both perfect competition and fully revealing prices ϕ are obtained because χχˆ → 1 and 1− ˆ → 0 and ϕˆ → 1 simultaneously. 1−ϕˆ → 1 as both χ 25

Intuitively, prices can be made more informative by making noise trading σZ small, and making the market simultaneously more competitive requires M to grow large 1 faster to compensate for the effect from increased price informativeness (M = χˆ (1− ϕˆ) ). This implies as the price becomes more informative, increasingly more traders must share and compete over the same information in order for prices to be fully revealing. The fact that such a strong condition is necessary to replicate the results of competitive models shows that perfect competition is not an adequate approximation for markets with a large number of traders and, thus, strategic trading motives must be taken into account when traders have private information.

5 Conclusion By examining the comparative statics properties of an equilibrium in demand schedules, we have found that a strategic equilibrium does not approach a rational expectations equilibrium as the market becomes large. Perfect competition and fully revealing prices cannot be achieved simultaneously. The result that traders choose not to trade when endowment shocks are sufficiently small indicates that an equilibrium in demand schedules does not efficiently internalize gains from trade. It is an interesting question for future research whether other trading mechanisms can achieve greater gains from trade than the single-price double auction analyzed in this paper.

26

References Akerlof, George A. 1970. “The Market for ‘Lemons’: Quality Uncertainty and the Market Mechanism.” Quarterly Journal of Economics, 84(3): 488–500. Bergemann, Dirk, Tibor Heumann, and Stephen Morris. 2015. “Information and Market Power.” Cowles Foundation Discussion Paper No. 2017, available at https:// ssrn.com/abstract=2639692 or http://dx.doi.org/10.2139/ssrn.2639692. Diamond, Douglas W, and Robert E Verrecchia. 1981. “Information Aggregation in a Noisy Rational Expectations Economy.” Journal of Financial Economics, 9(3): 221– 235. Dow, James, Vicente Madrigal, and Sergio Ribeira da Costa Werlang. 1990. “Preferences, Common Knowledge, and Speculative Trade.” London Business School, available at http://bibliotecadigital.fgv.br/dspace/bitstream/handle/10438/ 417/000075109.pdf?sequence=1. Duffie, Darrell, and Haoxiang Zhu. Forthcoming. “Size Discovery.” Review of Financial Studies, Stanford University Graduate School of Business Research Paper No. 15–56, available at https://ssrn.com/abstract=2682668 or http://dx.doi. org/10.2139/ssrn.2682668. Glode, Vincent, and Christian Opp. 2016. “Asymmetric Information and Intermediation Chains.” American Economic Review, 106(9): 2699–2721. Grossman, Sanford J., and Joseph E. Stiglitz. 1980. “On the Impossibility of Informationally Efficient Markets.” American Economic Review, 70(3): 393–408. Hellwig, Martin F. 1980. “On the Aggregation of Information in Competitive Markets.” Journal of Economic Theory, 22(3): 393–408. Kremer, Ilan. 2002. “Information Aggregation in Common Value Auctions.” Econometrica, 70(4): 1675–1682. Kyle, Albert S. 1985. “Continuous Auctions and Insider Trading.” Econometrica, 53(6): 1315–1335.

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Kyle, Albert S. 1989. “Informed Speculation with Imperfect Competition.” Review of Economic Studies, 56(3): 317–355. Kyle, Albert S., and Jeongmin Lee. 2016. “Information Aggregation with Symmetry.” Working Paper, available at http://financetheory.org/wp-content/uploads/ 2016/11/RM15.Lee_.pdf. Kyle, Albert S., Anna A. Obizhaeva, and Yajun Wang. 2016. “Smooth Trading with Overconfidence and Market Power.” Working Paper, University of Maryland, available at http://ssrn.com/abstract=2423207. Liu, Hong, and Yajun Wang. 2016. “Market Making With Asymmetric Information and Inventory Risk.” Journal of Economic Theory, 163: 73–109. Malamud, Semyon, and Marzena Rostek. 2016. “Decentralized Exchange.” Working Paper, Swiss Finance Institute, available at http://sfi.epfl.ch/files/content/ sites/sfi/files/users/192820/public/DE.pdf. Milgrom, Paul, and Nancy Stokey. 1982. “Information, Trade and Common Knowledge.” Journal of Economic Theory, 26(1): 17–27. Milgrom, Paul R. 1981. “Rational Expectations, Information Acquisition, and Competitive Bidding.” Econometrica, 49(4): 921–943. Pesendorfer, Wolfgang, and Jeroen M Swinkels. 1997. “The Loser’s Curse and Information Aggregation in Common Value Auctions.” Econometrica, 65(6): 1247–1281. Reny, Philip J, and Motty Perry. 2006. “Toward a Strategic Foundation for Rational Expectations Equilibrium.” Econometrica, 74(5): 1231–1269. Rostek, Marzena, and Marek Weretka. 2012. “Price Inference in Small Markets.” Econometrica, 80(2): 687–711. Rostek, Marzena, and Marek Weretka. 2015. “Information and Strategic Behavior.” Journal of Economic Theory, 158(Part B): 536 – 557. Symposium on Information, Coordination, and Market Frictions. Tirole, Jean. 1982. “On the Possibility of Speculation Under Rational Expectations.” Econometrica, 50(5): 1163–1181. 28

Vives, Xavier. 2011. “Strategic Supply Function Competition With Private Information.” Econometrica, 79(6): 1919–1966. Wilson, Robert. 1977. “A Bidding Model of Perfect Competition.” Review of Economic Studies, 44(3): 511–518.

29

A

Proofs As discussed, we assume πC = 0 without loss of generality. Since

Proof of Lemma 1.

πI , 0 and (N − 1) τI , 0, there is information asymmetry and the equilibrium price is potentially informative about the signals of traders in other groups. Trader (m, n ) can learn from the intercept of the residual supply schedule p m, n in (8). (M N − 1) πP

p m, n

σV

X

=

(m ′, n ′ ),(m, n )

(πI in ′ − πS ρσV sn ′ ) + ρσV z .

(44)

When M , 1, the value of p m, n already includes private information in and endowment sn because of trading of other traders in his group. By subtracting the overlapping information and endowments, we get (M N − 1) πP

p m, n

σV

− (M − 1) πI in + (M − 1) πS ρσV sn X X =M πI in ′ − M πS ρσV sn ′ + ρσV z . n ′ ,n

(45)

n ′ ,n

Dividing both sides by M (N − 1) πI yields p m, n (M N − 1) (M − 1) (M − 1) in + ρσV sn − M (N − 1) πI /πP σV M (N − 1) M (N − 1) πI /πS P P ′ ρσV sn ′ ρσV z n ′ ,n in ′ = − n ,n + . N −1 (N − 1) πI /πS M (N − 1) πI

Since v , e n , and

P

n ′ ,n

en ′

N −1

−

πS πI

is straightforward to show that

P

n ′ ,n

ρσV sn ′

(46)

Vz + M (ρσ are independently distributed, it N −1) πI

N −1

) p m, n (M N − 1) (M − 1) (M − 1) τ = σV · var v in , M (N − 1) πI /πP σV − M (N − 1) in + M (N − 1) πI /πS ρσV sn (P ) P πS n ′,n ρσV sn ′ ρσV z n ′ ,n e n ′ −1 = 1 + τI + τI var − + . N −1 πI N −1 M (N − 1) πI (47) ∗

2

−1

(

Writing τ∗ in the form of (10) implies

ρσV σS 1 =1+ ϕ πI /πS

!2

1 ρσV σZ + N −1 M πI

30

!2

.

(48)

The conditional expectation is a weighted average of two signals. v

) p m, n (M − 1) (M N − 1) (M − 1) in , E − in + ρσV sn σV M (N − 1) πI /πP σV M (N − 1) M (N − 1) πI /πS ! √ √ τI ϕ τI (M N − 1) p m, n (M − 1) (M − 1) = ∗ in + ∗ − ρσV sn in + τ τ M πI /πP σV M M πI /πS ! √ ! τI πP p m, n πS (N − 1) ϕπP p m, n M −1 = ∗ in + − in − − ρσV sn ϕ . τ πI σV πI σV πI M (49) (

If (N − 1) τI πI = 0, there is no learning from the price, and the conditional expectation

is simply

(

v n

)

√

τI in , sn, p m, n = ∗ in , σV τ which is consistent with (49) when ϕ = 0 is substituted. E

(50)

Proof of Lemma 2. Substituting the conditional expectation (13) into the optimal demand ((19)) allows the optimal quantity demanded xm, n to be written as 1/2

∗

1/2 πP

(1 + 2 χ) M ρσV xm, n = τI in − τ − (N − 1) ϕτI −

(M − 1) ϕτI1/2 M

πI

!

p m, n

σV

− ρσV sn

! πP p m, n πS ρσV sn . in − − πI σV πI

(51)

Substituting the residual supply curve (7) allows the optimal quantity demanded xm, n in (51) to be implemented with a demand schedule X m, n given by ! ϕτI1/2 π p P 1/2 1/2 ∗ *1 + χ + + M ρσV X m, n (p | in , sn ) = τ in − τ − (N − 1) ϕτ − ρσV sn I I π σ M 2 πI I V , ! (M − 1) ϕτI1/2 πP p πS − in − − ρσV sn . M πI σV πI (52) In moving from the optimal quantity on the residual supply schedule (51) to the demand schedule (52), we have shown that a trader in a symmetric linear equilibrium can implement a strategy which picks the best point on the linear supply schedule by submitting a demand schedule which is a function of price and is measurable with 31

respect to his private information and endowment shock. Proof of Theorem 1.

Collecting terms in the demand schedule (21) yields

! ϕτ1/2 *1 + χ + I +M ρσV X m, n (p | in , sn ) = 1 − (M − 1) ϕ τ1/2 in I M M 2 πI , ϕτI1/2 πP (M − 1) ϕτI1/2 πS 1 + p ∗ * * + ρσV sn . − τ − N − − 1− π M σ M πI I V , , -

(53)

Now equating the coefficients in the best response demand schedule produces the following three equations: τ∗ − ϕτI1/2 N − M1 ππPI πP = (54) 1/2 , −1) ϕ πI τI 1 − (M M (M −1) ϕτ 1/2

πS I 1− πS M πI = , 1/2 ( M −1) ϕ πI τI 1− M 1/2 −1) ϕ 1 − (MM τI πI = . 1/2 ϕτI 1 + χ + M 2π M

(55)

(56)

I

Combining with (16), these equations can be solved as functions of M , N , τI , and ϕ:

πI =

πS = πP =

(M N − 2 − (M N + N − 2) ϕ) τI1/2 , M (M N − 1)

1

π 1/2 I

τI

τ∗

=

π = 1/2 I

(1 + (N − 1) ϕ) τI

and

χ=

(M N − 2 − (M N + N − 2) ϕ) , M (M N − 1) (M N − 2 − (M N + N − 2) ϕ) τ∗ , M (M N − 1) (1 + (N − 1) ϕ)

1 + (N − 1) ϕ . M N − 2 − (M N + N − 2) ϕ

(57)

(58) (59)

(60)

Substituting πI , πP , πS , and χ into (21) yields (24). Substituting (24) into the market clearing condition (3) and rearranging yields (25). To determine the value of ϕ, first consider the case when (N − 1) τ , 0. substituting 32

(58) and (57) into (12) yields 1 − ϕ ( ρσV σS )2 ( ρσV σZ )2 / (N − 1) = + 2 . N −2 M N +N −2 ϕ τI τI M − ϕ M N −1 M N −1

(61)

To exclude trivial no-trade equilibrium, we require (M N − 1) πP , 0, that is, M N − 2 − (M N + N − 2) ϕ , 0.

(62)

The second-order condition ( χ > − 12 ) combined with no trivial no-trade equilibrium implies that MN − 2 M ϕ< ϕ> or . (63) MN + N − 2 M −1 The second inequality is ruled out because ϕ ≤ 1. If (N − 1) τ = 0, then ϕ = 0. This satisfies the second-order condition because M N > 2. Proof of Corollary 1. Proving this corollary is accomplished by analyzing the two equations (22) and (23) determining the endogenous parameter ϕ. First we prove if part. If M N > 2 and (N − 1) τI = 0, ϕ = 0 and this satisfies the second order condition (23). If M N > 2 and (N − 1) τI , 0, ϕ is determined by (22). Rewrite (22) as L ( ϕ) = R ( ϕ),

(64)

where we define L (ϕ) and R ( ϕ) by L ( ϕ) :=

1 − ϕ ( ρσV σS )2 − ϕ τI

(65)

and ( ρσV σZ )2 / (N − 1) R ( ϕ) := 2 . N −2 M N +N −2 τI M − ϕ M N −1 M N −1

(66)

If σZ = 0, (22) reduces to L (ϕ) = 0 with a unique solution. The solution satisfies 23 if and only if (29) holds. If σZ > 0, R (ϕ) is strictly and monotonically increasing for all ϕ in 0, MMN N+N−2−2 , with R (0) < ∞ and R (ϕ) → ∞ as ϕ → MMN N+N−2−2 . L ( ϕ) is strictly decreasing everywhere, with L (ϕ) → ∞ as ϕ → 0 and L MMN N+N−2−2 < ∞. Therefore there always exists a unique solution to (64) satisfying 0 < ϕ < 33

M N −2 . M N +N −2

Now we prove only if part. It suffices to show that there exists no equilibrium when M N ≤ 2. If M N ≤ 2, (23) implies ϕ < 0, which cannot be satisfied because 0 ≤ ϕ ≤ 1. Proof of Corollary 2. Equation (32) is a direct implication of Corollary 1 and Theorem 1 when σZ2 = 0 is substituted into (22). This completes the proof. From (51), it follows that

Proof of Corollary 3.

BR

!

πP = πI

τ∗ −

(N −1) ϕτI1/2 πP πI

τI1/2 −

which simplifies to BR

This implies that

πP πI

√

! τI πP 1 = ∗ τ πI 1− 1−

1 M

ϕ

−

−

(M −1) ϕτI1/2 πP M πI

(M −1) ϕτI1/2

(67)

,

M

N − M1 ϕ

1− 1−

1

M

√

ϕ

τI πP τ∗ πI

!

(68)

that satisfies (35) solves the fixed point problem in the sense

πP πI

would be an equilibrium value if ϕ were an equilibrium value. Again from (51), it follows that (M −1) ϕτI1/2 πP M πI

τ∗ − (N − 1) ϕτI1/2 ππPI − BR (πP ) = ϕτI1/2 1 + χ + M 2π M

(69)

I

By substituting (35) and (16) into (69), we get

BR (πP ) =

(N − 1 )ϕ τ∗ 1 − 1+(N M−1) ϕ

. (N − 1 )ϕ 1 + χ + χ (1+(N M−1) ϕ) M

(70)

The resulting market noncompetitiveness BR ( χ) follows substituting (70) into the definition of χ (16): BR (τ ∗ ) BR ( χ ) = , (71) M (M N − 1) BR (πP ) which yields (36).

34

Proof of Theorem 2. For given k , economy Ek has an equilibrium because exogenous noise trading satisfies σZ2 (k ) > 0. In equilibrium, ϕ is determined as a solution to (22), or, equivalently, as a solution to (64). Therefore for all k , there exists unique ϕk which solves M N − 2 M N + N − 2 2 ( ρσV σZ (k ))2 = R ( ϕk ) · − ϕk , (N − 1) MN − 1 MN − 1

(72)

and satisfies the second-order condition (23). As k approaches infinity, the left hand side of (72) approaches zero. Then possible solutions to (72) solve either R (ϕk ) → 0 or

ϕk →

MN − 2

MN + N − 2

.

(73)

Consider a candidate solution ϕcandidate that solves R ( ϕcandidate) = 0. Since an equilibrium does not exist in economy E and more specifically (29) does not hold, ϕcandidate

does not satisfy the second-order condition (23). Therefore, we obtain (37). Substitut ing (37) into (26) yields (38).

35

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